TY - JOUR

T1 - Potentially GL 2 -type galois representations associated to noncongruence modular forms

AU - Winnie Li, Wen Ching

AU - Liu, Tong

AU - Long, Ling

N1 - Funding Information:
The first author was supported in part by NSF grant DMS #1101368, Simons Foundation grant # 355798, and MOST grant 105-2811-M-001-002. The second author was supported by NSF grant DMS #1406926. The third author was supported by NSF grants DMS #1303292 and #1602047. The authors are grateful to the anonymous referee for the valuable suggestions and careful reading of this paper. They would also like to thank Michael Larson and Chia-Fu Yu for enlightening conversations and Fang-Ting Tu for her computational assistance. Part of the paper was written when the first and third authors were research fellows at the Institute for Computational and Experimental Research in Mathematics (ICERM) in fall 2015. They would like to thank ICERM for its hospitality. The first author would also like to thank the Institute of Mathematics, Academia Sinica in Taiwan for its support and hospitality in spring 2016 when she worked on this paper.
Funding Information:
Received by the editors April 5, 2017, and, in revised form, July 26, 2017, and August 8, 2017. 2010 Mathematics Subject Classification. Primary 11F11, 11F80. Key words and phrases. Galois representations, noncongruence modular forms, automorphy. The first author was supported in part by NSF grant DMS #1101368, Simons Foundation grant # 355798, and MOST grant 105-2811-M-001-002. The second author was supported by NSF grant DMS #1406926. The third author was supported by NSF grants DMS #1303292 and #1602047.
Publisher Copyright:
© 2019 American Mathematical Society.

PY - 2019

Y1 - 2019

N2 - In this paper, we consider representations of the absolute Galois group Gal(ℚ/ℚ) attached to modular forms for noncongruence subgroups of SL 2 (ℤ). When the underlying modular curves have a model over ℚ, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49-77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over ℚ. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL 2 -type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.

AB - In this paper, we consider representations of the absolute Galois group Gal(ℚ/ℚ) attached to modular forms for noncongruence subgroups of SL 2 (ℤ). When the underlying modular curves have a model over ℚ, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49-77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over ℚ. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL 2 -type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.

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U2 - 10.1090/tran/7364

DO - 10.1090/tran/7364

M3 - Article

AN - SCOPUS:85065333934

VL - 371

SP - 5341

EP - 5377

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -